Integer multiplication & division assessment
Lesson 24 of 24
Objective: SWBAT create examples of integer addition and subtraction problems to fit given criteria.
This warm up helps students learn strategies to help them make sense of the problem and persist in solving it. (mp1)
Students may have some trouble getting started. I would circulate to see if any ideas surface and share them with the class. For example, students might start making up some numbers and trying to subtract them and I would let the class know when someone started doing this. This is an important part of making sense of a problem. It leads to questions like "did we learn something from trying this?" or "what did we learn from trying this?" and "what could we try next"... "...instead?"
They begin to develop thinking like "Well, I tried this and it didn't work because of this, so I am going to try this different thing next...", etc. This helps build their confidence when they approach a problem they are initially completely confused by. It shifts their attitude from one of giving up when they don't immediately see a clear path to one of continually refining their approach until it leads them to a path to a solution, which is what they will have to do in life all the time.
This Warmup integer operations.docx tells students that Priscilla looked at an integer subtraction problem and said "You can find the answer by doubling the first number." It asks them what the problem might have been.
I expect students to start trying random subtraction problems. Students may say that they tried some subtraction, but the answer wasn't twice the first number, so it didn't tell us anything. I would suggest they try to do it Priscilla's way and just double the first number before figuring out what was subtracted. For example, I might ask a student to think of the first number and then figure out what the answer would be by doubling it, then figuring out what was subtracted. I may do this with the whole class if they are at about the same pace, or with small groups.
For example 10 - ____ = 20 (doubled 10) or -5 - ____ = -10 (doubled -5)
Students may just stick to positive numbers for the first number, which is fine, because they will still see the pattern. I can still ask "what would you have to subtract from positive 10 to get 20?" As students are working with this I would circulate again and see if anyone is trying to first figure out what to ADD to 10 in order to get 20. When I see this I bring it to the attention of the class by saying "here is something Jayden is trying....10 + ____ = 20, is this an easier problem?" They can all answer this. Then I would ask them how this helps us figure out what could be taken away instead of added. I would encourage them to think about what we learned about hot and cold cubes in earlier lessons (Mathmaster Chef series). Some students should be able to figure out that if +10 is added we could subtract -10 instead. If they don't see this I would remind them that we changed this to addition because it is easier, but what would the subtraction have been before we changed it to 10 + 10 = 20? That should lead them to 10 - (-10) = 20.
Now they should be able to generate several different examples and show that they work.
Assessment & homework
This assessment multiplication and division.docx has problems that require students to multiply and divide with two factors and also fill in the missing factor in multiplication problems with two and three factors. I grade and return the tests during class. I don't put a score or a number at the top, but only mark wrong answers. I have noticed that when I just do this they are more likely to make corrections, ask for help from their peers or from me for the ones they got wrong, and take a retake test. When I put a score at the top they turn it over and don't even look at the one's they got wrong. It's as if I have passed judgement and all learning has halted.
They know that they can make corrections, show them to me and explain what they did to correct it and then take a retest later that week.
Some mistakes that I expect are in the three factor multiplication. They sometimes think that if all the factors are negative the product will be positive. Some students, especially those who don't know their multiplication facts well, spend too much time on getting the number and then forget to figure out the sign, so leave it off. A small number of students may just add them all. They usually get this feedback from their math family group members when they compare their tests and I ask them to explain it to me when they give me the corrections. If a student who didn't do well does not turn their corrections in to me I will follow up with them privately.
Their homework negative 1 2 3 4 problems.docx will take them several days to figure out and complete. They are asked to make all the numbers 1 through 5 using only the numbers -1, -2, -3, -4 exactly one time using any operation or combination of operations. This is similar to a problem they did earlier in the year (4 Fours & The Power of Exponents). They will take it out each day and ask me questions and get feedback from me. The two key points are that all four negative numbers need to be used in each problem and that if they are going to make positive numbers out of negative numbers just adding won't work. This usually takes a couple days to clarify these two points. Some students finish and turn it in earlier and I point out to the class that these students ended up doing a lot of subtraction and multiplying to make positives out of negatives.